A Course in Functional Analysis

Cover
Springer Science & Business Media, 25.01.1994 - 400 Seiten
Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts.
 

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Inhalt

Hilbert Spaces
1
2 Orthogonality
7
3 The Riesz Representation Theorem
11
4 Orthonormal Sets of Vectors and Bases
14
5 Isomorphic Hilbert Spaces and the Fourier Transform for the Circle
19
6 The Direct Sum of Hilbert Spaces
23
Operators on Hilbert Space
26
2 The Adjoint of an Operator
31
3 Compact Operators
173
4 Invariant Subspaces
178
5 Weakly Compact Operators
183
Banach Algebras and Spectral Theory for Operators on a Banach Space
187
2 Ideals and Quotients
191
3 The Spectrum
195
4 The Riesz Functional Calculus
199
5 Dependence of the Spectrum on the Algebra
205

3 Projections and Idempotents Invariant and Reducing Subspaces
36
4 Compact Operators
41
5 The Diagonalization of Compact SelfAdjoint Operators
46
SturmLiouville Systems
49
7 The Spectral Theorem and Functional Calculus for Compact Normal Operators
54
8 Unitary Equivalence for Compact Normal Operators
60
Banach Spaces
63
2 Linear Operators on Normed Spaces
67
3 Finite Dimensional Normed Spaces
69
4 Quotients and Products of Normed Spaces
70
5 Linear Functionals
73
6 The HahnBanach Theorem
77
Banach Limits
82
Runges Theorem
83
Ordered Vector Spaces
86
10 The Dual of a Quotient Space and a Subspace
88
11 Reflexive Spaces
89
12 The Open Mapping and Closed Graph Theorems
90
13 Complemented Subspaces of a Banach Space
93
14 The Principle of Uniform Boundedness
95
Locally Convex Spaces
99
2 Metrizable and Normable Locally Convex Spaces
105
3 Some Geometric Consequences of the HahnBanach Theorem
108
4 Some Examples of the Dual Space of a Locally Convex Space
114
5 Inductive Limits and the Space of Distributions
116
Weak Topologies
124
2 The Dual of a Subspace and a Quotient Space
128
3 Alaoglus Theorem
130
4 Reflexivity Revisited
131
5 Separability and Metrizability
134
The StoneCech Compactification
137
7 The KreinMilman Theorem
141
The StoneWeierstrass Theorem
145
9 The Schauder Fixed Point Theorem
149
10 The RyllNardzewski Fixed Point Theorem
151
Haar Measure on a Compact Group
154
12 The KreinSmulian Theorem
159
13 Weak Compactness
163
Linear Operators on a Banach Space
166
2 The BanachStone Theorem
171
6 The Spectrum of a Linear Operator
208
7 The Spectral Theory of a Compact Operator
214
8 Abelian Banach Algebras
218
9 The Group Algebra of a Locally Compact Abelian Group
223
CAlgebras
232
2 Abelian CAlgebras and the Functional Calculus in CAlgebras
236
3 The Positive Elements in a CAlgebra
240
4 Ideals and Quotients of CAlgebras
245
5 Representations of CAlgebras and the GelfandNaimarkSegal Construction
248
Normal Operators on Hilbert Space
255
2 The Spectral Theorem
262
3 StarCyclic Normal Operators
268
4 Some Applications of the Spectral Theorem
271
5 Topologies on B H
274
6 Commuting Operators
276
7 Abelian von Neumann Algebras
281
The Conclusion of the Saga
285
9 Invariant Subspaces for Normal Operators
290
A Complete Set of Unitary Invariants
293
Unbounded Operators
303
2 Symmetric and SelfAdjoint Operators
308
3 The Cayley Transform
316
4 Unbounded Normal Operators and the Spectral Theorem
319
5 Stones Theorem
327
6 The Fourier Transform and Differentiation
334
7 Moments
343
Fredholm Theory
347
2 Fredholm Operators
349
3 Fredholm Theory
352
4 The Essential Spectrum
358
5 The Components of J F
362
6 A Finer Analysis of the Spectrum
363
Preliminaries
369
2 Topology
371
The Dual of Lp𝜇
375
The Dual of C₀X
378
Bibliography
384
List of Symbols
391
Index
395
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